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Deformations of normal vector fields and the generalized Minkowski problem
Author:
Herman Gluck
Journal:
Bull. Amer. Math. Soc. 77 (1971), 1106-1110
MSC (1970):
Primary 53C45, 53C20, 57D25; Secondary 57D50
MathSciNet review:
0284955
Full-text PDF
References |
Similar Articles |
Additional Information
- 1.
A. D. Alexandrov, Zur Theorie gemischter Volumina konvexer Körper. I, II, III, Mat. Sb. 2 (1937), 947-972; 2 (1937), 1205-1238; 3 (1938).
- 2.
A. D. Alexandrov, Von der Flächenfunktion eines konvexen Körpers, Mat. Sb. 6 (48) (1939), 167.
- 3.
A.
Alexandroff, Smoothness of the convex surface of bounded Gaussian
curvature, C. R. (Doklady) Acad. Sci. URSS (N.S.) 36
(1942), 195–199. MR 0007626
(4,169d)
- 4.
Melvyn
S. Berger, Riemannian structures of prescribed Gaussian curvature
for compact 2-manifolds, J. Differential Geometry 5
(1971), 325–332. MR 0295261
(45 #4329)
- 5.
W. Blaschke, Vorlesungen über differential Geometrie, Berlin, 1930; reprint, Chelsea, New York, 1967.
- 6.
T.
Bonnesen and W.
Fenchel, Theorie der konvexen Körper, Springer-Verlag,
Berlin, 1974 (German). Berichtigter Reprint. MR 0344997
(49 #9736)
- 7.
Herbert
Busemann, Convex surfaces, Interscience Tracts in Pure and
Applied Mathematics, no. 6, Interscience Publishers, Inc., New York, 1958.
MR
0105155 (21 #3900)
- 8.
Samuel
Eilenberg and Norman
Steenrod, Foundations of algebraic topology, Princeton
University Press, Princeton, New Jersey, 1952. MR 0050886
(14,398b)
- 9.
Herman
Gluck, The generalized Minkowski problem in differential geometry
in the large, Ann. of Math. (2) 96 (1972),
245–276. MR 0309021
(46 #8132)
- 10.
Herman
Gluck, The converse to the four vertex theorem, Enseignement
Math. (2) 17 (1971), 295–309. MR 0344998
(49 #9737)
- 11.
Jerry
L. Kazdan and F.
W. Warner, Integrability conditions for
Δ𝑢=𝑘-𝐾𝑒^{𝛼𝑢} with
applications to Riemannian geometry, Bull.
Amer. Math. Soc. 77
(1971), 819–823. MR 0282314
(43 #8026), http://dx.doi.org/10.1090/S0002-9904-1971-12818-5
- 12.
H. Lewy, On the existence of a closed surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci. U.S.A. 24 (1938), 104-106.
- 13.
H. Minkowski, Allgemeine Lehrsätze über konvexer Polyeder, Nachr. Ges. Wiss. Göttingen 1897, 198-219.
- 14.
Hermann
Minkowski, Volumen und Oberfläche, Math. Ann.
57 (1903), no. 4, 447–495 (German). MR
1511220, http://dx.doi.org/10.1007/BF01445180
- 15.
J. Moser (to appear).
- 16.
Louis
Nirenberg, The Weyl and Minkowski problems in differential geometry
in the large, Comm. Pure Appl. Math. 6 (1953),
337–394. MR 0058265
(15,347b)
- 17.
A.
V. Pogorelov, Regularity of a convex surface with given Gaussian
curvature, Mat. Sbornik N.S. 31(73) (1952),
88–103 (Russian). MR 0052807
(14,679b)
- 1.
- A. D. Alexandrov, Zur Theorie gemischter Volumina konvexer Körper. I, II, III, Mat. Sb. 2 (1937), 947-972; 2 (1937), 1205-1238; 3 (1938).
- 2.
- A. D. Alexandrov, Von der Flächenfunktion eines konvexen Körpers, Mat. Sb. 6 (48) (1939), 167.
- 3.
- A. D. Alexandrov, Smoothness of the convex surface of bounded Gaussian curvature. C. R. (Dokl.) Acad. Sci. URSS 36 (1942), 195-199. MR 4, 69. MR 7626
- 4.
- M. Berger, On Riemannian structures of prescribed Gauss curvature for compact two-dimensional manifolds, J. Diff. Geometry 5 (1971) (to appear). MR 295261
- 5.
- W. Blaschke, Vorlesungen über differential Geometrie, Berlin, 1930; reprint, Chelsea, New York, 1967.
- 6.
- T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Julius Springer, Berlin, 1934. MR 344997
- 7.
- H. Busemann, Convex surfaces, Interscience Tracts in Pure and Appl. Math., no. 6, Interscience, New York, 1958. MR 21 #3900. MR 105155
- 8.
- S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton Univ. Press, Princeton, N. J., 1952. MR 14, 398. MR 50886
- 9.
- H. Gluck, The generalized Minkowski problem in differential geometry in the large, Ann. of Math, (to appear). MR 309021
- 10.
- H. Gluck, The converse to the Four Vertex Theorem, l'Enseignement Math. (to appear). MR 344998
- 11.
- J. Kazdan and F. Warner, Integrability conditions for $\Delta u=k-Ke\sp{\alpha u}$ with applications to Riemannian geometry, Bull. Amer. Math. Soc. 77 (1971), 819-823. MR 282314
- 12.
- H. Lewy, On the existence of a closed surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci. U.S.A. 24 (1938), 104-106.
- 13.
- H. Minkowski, Allgemeine Lehrsätze über konvexer Polyeder, Nachr. Ges. Wiss. Göttingen 1897, 198-219.
- 14.
- H. Minkowski, Volumen und Oberfläche, Math. Ann. 57 (1903), 447-495. MR 1511220
- 15.
- J. Moser (to appear).
- 16.
- L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337-394. MR 15, 347. MR 58265
- 17.
- A. V. Pogorelov, Regularity of a convex surface with given Gaussian curvature, Mat. Sb. 31 (73) (1952), 88-103. (Russian) MR 14, 679. MR 52807
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9904-1971-12885-9
PII:
S 0002-9904(1971)12885-9
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